Problem: Simplify the following expression: $ r = \dfrac{7}{4} + \dfrac{7q + 1}{-3q - 7} $
Solution: In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-3q - 7}{-3q - 7}$ $ \dfrac{7}{4} \times \dfrac{-3q - 7}{-3q - 7} = \dfrac{-21q - 49}{-12q - 28} $ Multiply the second expression by $\dfrac{4}{4}$ $ \dfrac{7q + 1}{-3q - 7} \times \dfrac{4}{4} = \dfrac{28q + 4}{-12q - 28} $ Therefore $ r = \dfrac{-21q - 49}{-12q - 28} + \dfrac{28q + 4}{-12q - 28} $ Now the expressions have the same denominator we can simply add the numerators: $r = \dfrac{-21q - 49 + 28q + 4}{-12q - 28} $ $r = \dfrac{7q - 45}{-12q - 28}$ Simplify the expression by dividing the numerator and denominator by -1: $r = \dfrac{-7q + 45}{12q + 28}$